Question: The equation of hyperbola $H$ is $\dfrac {(y+5)^{2}}{9}-\dfrac {(x+5)^{2}}{49} = 1$. What are the asymptotes?
We want to rewrite the equation in terms of $y$ , so start off by moving the $y$ terms to one side: $\dfrac {(y+5)^{2}}{9} = 1 + \dfrac {(x+5)^{2}}{49}$ Multiply both sides of the equation by $9$ $(y+5)^{2} = { 9 + \dfrac{ (x+5)^{2} \cdot 9 }{49}}$ Take the square root of both sides. $\sqrt{(y+5)^{2}} = \pm \sqrt { 9 + \dfrac{ (x+5)^{2} \cdot 9 }{49}}$ $ y + 5 = \pm \sqrt { 9 + \dfrac{ (x+5)^{2} \cdot 9 }{49}}$ As $x$ approaches positive or negative infinity, the constant term in the square root matters less and less, so we can just ignore it. $y + 5 \approx \pm \sqrt {\dfrac{ (x+5)^{2} \cdot 9 }{49}}$ $y + 5 \approx \pm \left(\dfrac{3 \cdot (x + 5)}{7}\right)$ Subtract $5$ from both sides and rewrite as an equality in terms of $y$ to get the equation of the asymptotes: $y = \pm \dfrac{3}{7}(x + 5) -5$